Illinois Journal of Mathematics

Naimark’s problem for graph $C^{*}$-algebras

Nishant Suri and Mark Tomforde

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Naimark’s problem asks whether a $C^{*}$-algebra that has only one irreducible $*$-representation up to unitary equivalence is isomorphic to the $C^{*}$-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable $C^{*}$-algebras and Type I $C^{*}$-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a $C^{*}$-algebra with $\aleph_{1}$ generators that is a counterexample to Naimark’s Problem. More precisely, they showed that the statement “There exists a counterexample to Naimark’s Problem that is generated by $\aleph_{1}$ elements.” is independent of the axioms of ZFC. Whether Naimark’s problem itself is independent of ZFC remains unknown. In this paper, we examine Naimark’s problem in the setting of graph $C^{*}$-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph $C^{*}$-algebras as well as for $C^{*}$-algebras of graphs in which each vertex emits a countable number of edges.

Article information

Illinois J. Math., Volume 61, Number 3-4 (2017), 479-495.

Received: 21 September 2017
Revised: 14 May 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]


Suri, Nishant; Tomforde, Mark. Naimark’s problem for graph $C^{*}$-algebras. Illinois J. Math. 61 (2017), no. 3-4, 479--495. doi:10.1215/ijm/1534924836.

Export citation


  • C. Akemann and N. Weaver, Consistency of a counterexample to Naimark's problem, Proc. Natl. Acad. Sci. USA 101 (2004), no. 20, 7522–7525.
  • T. Bates, J. H. Hong, I. Raeburn and W. Szymański, The ideal structure of the $C^{*}$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176.
  • J. Dixmier, Sur les $C^{*}$-algébres (French), Bull. Soc. Math. France 88 (1960), 95–112.
  • J. M. G. Fell, $C^{*}$-Algebras with smooth dual, Illinois J. Math. 4 (1960), 221–230.
  • J. Glimm, Type I $C^{*}$-algebras, Ann. of Math. (2) 73 (1961), 572–612.
  • I. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219–255.
  • P. Muhly and M. Tomforde, Adding tails to $C^{*}$-correspondences, Doc. Math. 9 (2004), 79–106.
  • M. A. Naimark, Rings with involutions, Uspekhi Mat. Nauk 3 (1948), no. 5(27), 52–145. (Russian).
  • M. A. Naimark, On a problem of the theory of rings with involution, Uspekhi Mat. Nauk 6 (1951), no. 6(46), 160–164. (Russian).
  • A. L. T. Paterson, Graph inverse semigroups, groupoids and their $C^{*}$-algebras, J. Operator Theory 48 (2002), 645–662.
  • G. Pedersen, $C^{*}$-Algebras and their automorphism groups, Academic Press Inc., New York, 1979.
  • I. Raeburn and W. Szymański, Cuntz–Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59.
  • I. Raeburn and D. P. Williams, Morita equivalence and continuous-trace $C^{*}$-algebras, Math. Surveys & Monographs, vol. 60, Amer. Math. Soc., Providence, 1998.
  • A. Rosenberg, The number of irreducible representations of simple rings with no minimal ideals, Amer. J. Math. 75 (1953), 523–530.
  • W. Szymański, Simplicity of Cuntz–Krieger algebras of infinite matrices, Pacific J. Math. 199 (2001), no. 1, 249–256.